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(II)
Luis Gonzalez-Mestres
To cite this version:
Luis Gonzalez-Mestres. CMB B-modes, spinorial space-time and Pre-Big Bang (II). 2014. �hal-
01037854�
CMB B -modes, spinorial space-time and Pre-Big Bang (II)
Luis Gonzalez-Mestres
∗Megatrend Cosmology Laboratory, Megatrend University, Belgrade and Paris Goce Delceva 8, 11070 Novi Beograd, Serbia
The BICEP2 collaboration reported recently a B-mode polarization of the cosmic microwave background (CMB) radiation inconsistent with the null hypothesis at a significance of > 5 σ.
This result has been often interpreted as a signature of primordial gravitational waves from cosmic inflation, even if actually polarized dust emission may be at the origin of such a signal.
Even assuming that part of this CMB B-mode polarization really corresponds to the early Universe dynamics, its interpretation in terms of inflation and primordial gravitational waves is not the only possible one. Alternative cosmologies such as pre-Big Bang patterns and the spinorial space-time (SST) we introduced in 1996-97 can naturally account for such CMB B- modes. In particular, the SST automatically generates a privileged space direction (PSD) whose existence may have been confirmed by Planck data. If such a PSD exists, it seems normal to infer that vector perturbations have been present in the early Universe leading to CMBB-modes in suitable cosmological patterns. Inflation would not be required to explain the BICEP2 result assuming it really contains a primordial signal. More generally, pre-Big Bang cosmologies can also generate gravitational waves in the early Universe without any need for cosmic inflation.
We further discuss here possible alternatives to the inflationary interpretation of a primordial B-mode polarization of cosmic microwave background radiation.
1. Introduction
BICEP2 data [ 1, 2] may have provided a sig- nature of CMB
B-modes from the early uni- verse. However, this is not yet certain as the observed signal could actually be due to galac- tic dust effects [ 3, 4] even if the experimental program looks promising in all cases [ 5, 6].
Furthermore, if the experimental and phe- nomenological uncertainty remains, the situa- tion is not different from a theoretical point of view, as already pointed out in [ 7].
Assuming that some of the recent BICEP2 data really correspond to a primordial
B-mode polarization of cosmic microwave background radiation, the theoretical interpretation of such a signal would not yet be obvious. Alterna-
∗luis.gonzalez-mestres@megatrend.edu.rs at the Mega- trend Cosmology Laboratory
Luis.Gonzalez-Mestres@univ-savoie.fr at the Universit´e de Savoie
lgmsci@yahoo.fr, personal e-mail
tive cosmologies must be seriously considered [ 7, 8] and can even work more naturally than the standard Big Bang with cosmic inflation to explain the observed effect.
The BICEP2 result is often presented as a strong direct evidence for cosmic inflation and primordial gravitational waves. It is claimed that the
B-modes of CMB cannot be generatedprimordially by scalar (density) perturbations and that only gravitational waves (tensor per- turbations) originating from the inflationary ex- pansion of the Universe can produce them. Pos- sible vector perturbations (vorticity), that can in principle naturally generate such
B-modes,are not really considered within the conven- tional cosmological framework. But this kind of analysis applies only to cosmologies based on the standard Big Bang approach where inflation is a crucial ingredient.
The situation can be radically different in rea-
sonable alternative cosmologies. Pre-Big Bang
models [ 9, 10] do not in general require an in- flationary scenario and can efficiently produce primordial CMB
B-modes through vector per-turbations [ 7]. The spinorial space-time (SST) we suggested in 1996-97 [ 11, 12] automatically generates [ 13, 14] a privileged space direction (PSD) for each comoving observer. Then, the existence of CMB
B-modes is a natural conse- quence of this space anisotropy of geometric and cosmic origin.
Alternative cosmologies, including pre-Big Bang, are not ”exotic” and have not been ex- cluded by data. Physics beyond the Planck scale can be a natural extension of standard theories if quantum mechanics ceases to hold or undergoes modifications at very high ener- gies and very small distances [ 9, 15]. Similarly, the effective space-time structure can depend on the energy scale [ 16, 17]. In the case of SST, the existence of a privileged space direc- tion for each comoving observer, already com- patible with WMAP data [ 18], may have been confirmed by more recent Planck [ 19] results [ 20]. The PSD combined with parity viola- tion can potentially explain the observed CMB anisotropy [ 10, 13]. Pre-Big Bang models can naturally solve the horizon problem [ 16] and provide sensible alternatives to the inflation- ary description of the formation of conventional matter structure in our Universe. They can also generate primordial gravitational waves without the standard cosmic inflation, as shown in an approach based on an initial gravitational in- stanton at cosmic time
t= 0 [ 21, 22].
In this note, we further develop the analysis of [ 7] on possible alternatives to the inflation- ary interpretation of BICEP2 results assuming the
B-modes of CMB really correspond, at leastpartially, to a signal from the early Universe.
2. The spinorial space-time (SST)
It is well known that fermions do not feel space exactly in the same way as bosons and macroscopic objects. In particular, a 360 de- grees rotation changes the sign of a spin-1/2 wave function. To explore all possible conse-
quences of this property of particles with half- integer spin, we introduced [ 11, 12] a spino- rial SU(2) space-time with two complex coordi- nates replacing the four standard real ones. The properties of the SST, including its possible cos- mological implications, have been reminded and further studied in [ 9, 17] and in [ 13, 14].
2.1. SST basic structure
It turns out that the use of the SST instead of the conventional real space-time can have im- portant and connected implications for both the internal properties of standard elementary par- ticles and the very large scale structure of the Universe. The two domains are actually related through pre-Big Bang evolution, where the ulti- mate structure of matter and space-time is ex- pected to play a leading role and dominate the overall dynamics of the primordial Universe.
To a SU(2) spinor
ξdescribing cosmic space- time coordinates (two complex variables), it is possible to associate a positive SU(2) scalar
|ξ |2
=
ξ†ξ(the dagger stands for hermitic con- jugate). A definition of cosmic time (age of the Universe) can then be
t=
|ξ |with an associ- ated space given by the
S3hypersphere
|ξ |=
t. Other definitions of tin terms of
|ξ |(f.i.
t=
|ξ |2) lead to similar cosmological results as long as a single-valued function is used.
With the definition
t=
| ξ |, if ξ0is the ob- server position on the
| ξ |=
t0hypersphere, space translations inside this hypersphere are described by SU(2) transformations acting on the spinor space, i.e.
ξ=
U ξ0with:
U
=
exp(i/2
t−01 ~σ.~x)
≡U(~ x) (1) where
~σis the vector formed by the usual Pauli matrices and the vector
~x the spatial position (in time units, at that stage) of
ξwith respect to
ξ0at constant time
t0.
The origin of cosmic time, associated to the
beginning of the Universe, is naturally given
by the point
ξ= 0 where the initial space is
contracted to a single point. One then has an
expanding universe where cosmological comov-
ing frames correspond to straight lines going
through the time origin
ξ= 0. The SST ge- ometry naturally suggests the existence of a lo- cal privileged rest frame for each comoving ob- server, which is compatible with existing cosmo- logical observations.
As already pointed out in [ 11, 12], an at- tempt to associate to the cosmic spinor
ξreal cosmic space coordinates x
~cwriting x
~c=
ξ†~σξdoes not really lead to such coordinates. In- stead, one gets
|ξ |2times a unit vector defin- ing the local PSD. The standard space coordi- nates can only be defined from an origin
ξ0at the same cosmic time
t, as in (1). This situa-tion clearly illustrates the potential limitations of general relativity and standard cosmology.
Rather than an intrinsic fundamental property of space and time, conventional relativity would most likely be a low-energy symmetry of stan- dard matter similar to the well-known effective Lorentz-like symmetry of the kinematics of low- momentum phonons in a lattice [ 28, 29] where the speed of sound plays the role of the critical speed. The speed of light would then be the critical speed of a family of vacuum excitations (the standard particles) not directly related to an intrinsic space-time geometry.
Space rotations with respect to a fixed point
ξ0are given by SU(2) transformations acting on the spatial position vector
~x defined by (1).
A standard spatial rotation around
ξ0is now given by a SU(2) element
U(~ y) turning
U(~ x) into
U(~ y)
U(~ x)
U(~ y)
†. The vector
~y, related to
U(~ y) in a similar way to (1), provides the rotation axis and angle. If a spin-1/2 particle is present at the position
~x with an associated spinor
ξpdescribing its spin, then
ξptransforms into
ξp′=
U(~ y)
ξp.
2.2. Some direct consequences
It can be readily checked [ 9, 17] that the SST automatically generates two basic cosmological phenomena in a purely geometric way :
i) The standard relation between relative ve- locities and distances at cosmic scale, with a ratio
H(velocity/distance) equal to the inverse of the age of the Universe (H =
t−1).
ii) As previously stressed, a privileged space direction for each comoving observer.
Furthermore, space translations form a (non- abelian) compact group, contrary to standard space-time geometry.
The PSD associated to the space-time point
ξis defined by the linear combination of sigma matrices (with real coefficients) that leaves
ξin- variant. The space-time points on the trajec- tory generated by this sigma-like matrix satisfy the relation
ξ′=
exp(iφ)
ξwhere
φis real and
exp(iφ) is a complex phase factor. This defini- tion of the PSD is stable under SU(2) transfor- mations and comoving time evolution.
The existence of the PSD is a cosmological property specific to the spinorial structure of the cosmic space-time as ”seen” from the cosmic ori- gin
ξ= 0 (t = 0). The PSD is not apparent in the space-time geometry when standard space coordinates (the above
~x) are used, as these co- ordinates belong to a vector representation of SU(2) and SO(3). Thus, conventional cosmol- ogy based on the usual real space-time cannot in principle account for the PSD in a simple way. We expect bosons and usual macroscopic objects to be less directly concerned by PSD effects than the elementary fermions, the possi- ble ultimate constituents of matter and the very large scale structure of the Universe.
Contrary to the standard isotropic descrip-
tion of the early Universe, where only
E-modes associated to gradients are assumed to
be present in the CMB except for the
B-modes
due to inflationary gravitational waves, a cos-
mology based on the spinorial space-time nat-
urally leads to
B-modes generated by rotationsaround the privileged space direction and vec-
tor products by this direction. Then, cosmic
inflation and primordial gravitational waves are
no longer necessary to account for the primor-
dial CMB
B-modes that BICEP2 has possiblyobserved. On the contrary, such a result, to-
gether with recent Planck data, may have pro-
vided a signature of SST geometry or of some
other unconventional structure beyond the stan-
dard space-time and cosmology.
2.3. SST, conservation laws, causality The existence of a PSD implies a potential vi- olation of rotation invariance in Particle Physics that may invalidate the standard conservation law for angular momentum in phenomena sensi- tive to the PSD. However, such an effect can be very difficult to detect in Particle Physics exper- iments, as orbital angular momentum is defined using position and momentum operators that are vector representations of the space symme- try group. The internal structure of fermions would be sensitive to the PSD and potentially lead to some observable signatures.
As the time-dependent global size of the Uni- verse is part of the fundamental space-time ge- ometry, one can consider that energy conserva- tion does no longer follow as an exact basic law of Physics. Even if we expect the possible effects of energy non conservation due to the Universe expansion to be too small to be detected in lab- oratory experiments, the possible evolution of vacuum structure and particle properties at cos- mological scales must be carefully explored.
When using the SST geometry for conven- tional particles, it seems normal to describe the internal structure of standard elementary fermions (quarks and leptons) through a spino- rial wave function defined in a local SST whose origin lies at the particle space-time position.
Then, for a comoving particle at
ξ0, the local spinorial coordinates of a point
ξwould be given by the spinor
ξL=
ξ-
ξ0. Considering a wave function of the type Ψ(ξ
L) to describe the lep- ton and quark internal structure [ 9, 17] pro- vides an unconventional alternative to standard causality at very small distance and local time scales, as the values of
ξthus considered do not in general correspond to the same value of the cosmic time as
ξ0. At such scales, the notion of time itself should be renconsidered.
Assuming the internal wave function of a standard ”elementary” particle to be a SU(2) eigenstate, the allowed spin (spinorial angular momentum) values would be multiples of 1/2, including 0, 1/2, 1, 3/2 and 2 but also possibly higher spins contrary to standard assumptions.
All particles of standard physics can thus be generated by a spinorial wave function, and the existence of ”elementary” spin-3/2 parti- cles seems natural in such a pattern. As the Poincar´e group is no longer an exact symmetry in such an approach, an alternative to super- symmetry involving both space-time and inter- nal symmetries may thus emerge as a new (ap- proximate and broken) symmetry escaping stan- dard theorems [ 9, 17]. The subject clearly re- quires further exploration, including the exper- imental search for signatures of such ”elemen- tary” (similar to quarks and leptons) spin-3/2 particles and of possible spin-2 particles other than the graviton.
Similarly, the possible existence of ”elemen- tary” particles with spin larger than 2 can- not be excluded in such a pattern and deserves close theoretical and experimental investigation.
High spin elementary fields were already con- sidered in a different approach [ 23, 24]. An alternative to standard quantum field theory (SQFT) where the basic vacuum structure is not dominated by the usual field condensates and zero modes has been suggested in [ 30, 31]
and in [ 9, 17].
3. Pre-Big Bang, SST, PSD, super- bradyons,
B-modes...
In this and previous articles, we always con- sider pre-Big Bang scenarios that are not based on mere extrapolations from standard dynamics (including strings) to higher energies and lower distance scales. Our basic assumption about pre-Big Bang is that really new physics leads the Universe evolution at distance and time scales smaller than the Planck scale, and that the stan- dard principles of Physics (relativity, quantum mechanics...) cease to be valid at these scales or even at larger scales [ 25, 26]. New ultimate constituents of matter and a new space-time ge- ometry can then dominate this unconventional primordial phase of the history of the Universe.
Conventional gravity is not necessarily the ap-
propriate framework to understand the ultimate
origin of space and time.
In spite of its already important implications, the above described SST does not yet incorpo- rate space units, standard matter or even a def- inite vacuum structure. The size of the SST- based universe can be much larger than that of the conventional one. It may even happen that standard matter occupies only a small part of the SST, or that its nucleation has occurred in many independent regions. From a dynam- ical point of view, it seems normal to assume that the SST geometry is somehow in quasi- equilibrium with an underlying physical vac- uum, even if the SST structure and the evolu- tion of the Universe (the time-dependent radius) reflect by themselves the existence of dominant cosmic forces leading to this evolution in time.
As previously stressed, the notion of time itself deserves further thought [ 27].
If the vacuum is made of a fundamental mat- ter or pre-matter different from standard mat- ter and of which the conventional ”elementary”
particles are actually composite, the speed of light is not expected to be a fundamental crit- ical speed. The ultimate matter constituents can have a critical speed much faster than that of light just as the speed of light is much faster than that of sound [ 16, 28]. Then, it is not excluded that the ultimate fundamental objects (such as superbradyons [ 29]) exist in our Uni- verse as free superluminal particles. They can be remnants from the early Universe [ 11, 17]
and part of the dark matter [ 25, 30].
3.1. A new Friedmann-like equation As emphasized in [ 8, 10], the SST leads to a new approach to the role of space curvature in cosmology and to a new structure of Friedmann- like equations. In particular, the leading contri- bution to the square of the Lundmark-Lemaˆıtre- Hubble constant [ 9]
Hcomes from a curvature term equal to
t−2whose sign does not depend on the space curvature felt by standard matter.
In [ 8], the following Friedmann-like relation was considered :
H2
= 8π G ρ/3
−k R−2c2+
t−2+
K+ Λ
c2/3(2)
where
ρis the energy density associated to stan- dard matter,
cthe speed of light,
k R−2the curvature parameter,
Rthe present curvature distance scale of the Universe (the curvature ra- dius, and possibly the radius of the Universe, for
k= 1) and Λ a possible new version of the cos- mological constant decreasing like the matter density as the Universe expands. The new term
t−2, of cosmic geometric origin as suggested by the SST structure, dominates the large scale ex- pansion of the Universe.
Kis a correction term accounting in particular for:
- a possible small difference between the comoving frames of standard cosmology and those (pre-existing) obtained from the under- lying SST cosmic geometry;
- a reaction of the nucleated standard matter to the pre-existing expansion of the Universe led by the SST geometry [ 9, 10];
- vacuum inhomogeneities at cosmic scale and other non-standard effects.
A further modification of (2) accounting for phenomena related to the local privileged space direction in the explicit presence of matter and pre-matter should also be considered.
3.2. The superbradyon hypothesis
Superbradyons (superluminal preons) provide a simple explicit example of new ingredients that alternative cosmologies can naturally in- corporate in pre-Big Bang scenarios. Again, the existence of a privileged rest frame for each co- moving observer is naturally assumed [ 11, 29].
Superbradyons can be the basic constituents of the fundamental vacuum tacitly considered in the SST approach.
In a limit where the usual kinematical con- cepts would still make sense for such objects, a simple choice for the relation between their energy (E
s), momentum (p
s) and velocity (v
s) would be [ 29]:
Es
